| L(s) = 1 | + (1.41 − 2.44i)2-s + (−4.5 + 2.59i)3-s + (−3.99 − 6.92i)4-s + (−20.7 − 11.9i)5-s + 14.6i·6-s + (−47.1 − 13.1i)7-s − 22.6·8-s + (13.5 − 23.3i)9-s + (−58.6 + 33.8i)10-s + (−48.9 − 84.8i)11-s + (35.9 + 20.7i)12-s + 104. i·13-s + (−98.9 + 96.9i)14-s + 124.·15-s + (−32.0 + 55.4i)16-s + (93.2 − 53.8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.829 − 0.479i)5-s + 0.408i·6-s + (−0.963 − 0.268i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.586 + 0.338i)10-s + (−0.404 − 0.701i)11-s + (0.249 + 0.144i)12-s + 0.616i·13-s + (−0.505 + 0.494i)14-s + 0.553·15-s + (−0.125 + 0.216i)16-s + (0.322 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0366977 - 0.587133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0366977 - 0.587133i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 2.44i)T \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (47.1 + 13.1i)T \) |
| good | 5 | \( 1 + (20.7 + 11.9i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (48.9 + 84.8i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 104. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-93.2 + 53.8i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-33.2 - 19.1i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-510. + 884. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 621.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (1.31e3 - 759. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-281. + 486. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 1.02e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.38e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (3.41e3 + 1.97e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.09e3 + 1.89e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-2.54e3 + 1.46e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-576. - 332. i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.96e3 - 5.13e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.49e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-7.76e3 + 4.48e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-5.22e3 + 9.04e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.26e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (3.12e3 + 1.80e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 1.95e3iT - 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.67685531002169974126742736851, −13.20919818558962458433537048130, −12.28916386568937250424629118761, −11.21113520241450686464224636099, −10.05579358654567040896560908666, −8.616198391398499759997982866571, −6.62621496979631768334268126376, −4.88653222506638658314314640790, −3.43155639102114000304224415141, −0.35861722902111823352446220949,
3.39742501952175080326639774109, 5.36751440368906582812486783957, 6.84024858086050808473345619818, 7.81954727127925057209467155950, 9.701404869765280392685935704482, 11.27026949666567233932140266831, 12.47836595104777156382395614571, 13.33082551580988574949361930002, 15.05846113869720668370799804503, 15.61772213274061444139527336485
